4-7 Counting 189

Five different rules for finding total numbers of outcomes were given in this sec-tion. Although not all counting problems can be solved with one of these five rules,they do provide a strong foundation for many real and relevant applications.

Basic Skills and Concepts

Statistical Literacy and Critical Thinking1. Permutations and Combinations What is the basic difference between a situation re-quiring application of the permutations rule and one that requires the combinations rule?

2. Combination Lock The typical combination lock uses three numbers between 0 and 49,and they must be selected in the correct sequence. Given the way that these locks work, is thename of combination lock correct? Why or why not?

3. Trifecta In horse racing, a trifecta is a bet that the first three finishers in a race are selected,and they are selected in the correct order. Does a trifecta involve combinations or permuta-tions? Explain.

4. Quinela In horse racing, a quinela is a bet that the first two finishers in a race are selected, andthey can be selected in any order. Does a quinela involve combinations or permutations? Explain.

Calculating Factorials, Combinations, Permutations. In Exercises 512, evaluatethe given expressions and express all results using the usual format for writingnumbers (instead of scientific notation).5. Factorial Find the number of different ways that five test questions can be arranged in or-der by evaluating 5!.

6. Factorial Find the number of different ways that the nine players on a baseball team canline up for the National Anthem by evaluating 9!.

7. Blackjack In the game of blackjack played with one deck, a player is initially dealt twocards. Find the number of different two-card initial hands by evaluating C .

8. Card Playing Find the number of different possible five-card poker hands by evaluatingC .

9. Scheduling Routes A manager must select 5 delivery locations from 9 that are available.Find the number of different possible routes by evaluating P .

10. Scheduling Routes A political strategist must visit state capitols, but she has time tovisit only 3 of them. Find the number of different possible routes by evaluating P .

11. Virginia Lottery The Virginia Win for Life lottery game requires that you select the cor-rect 6 numbers between 1 and 42. Find the number of possible combinations by evaluating

C .

12. Trifecta Refer to Exercise 3. Find the number of different possible trifecta bets in a racewith ten horses by evaluating P .

Probability of Winning the Lottery. Because the California Fantasy 5 lottery iswon by selecting the correct five numbers (in any order) between 1 and 39, there are575,757 different 5-number combinations that could be played, and the probability

310

642

350

59

552

252

4-7

number of combinations. With n numbers available and with r numbersselected, the number of combinations is

With 1 winning combination and 22,957,480 different possible combinations, theprobability of winning the jackpot is .1>22,957,480

nCr =n !

(n - r)! r !=

53!

(53 - 6)! 6!= 22,957,480

= 6= 53

190 Chapter 4 Probability

of winning this lottery is In Exercises 1316, find the probability ofwinning the indicated lottery by buying one ticket. In each case, numbers selectedare different and order does not matter. Express the result as a fraction.13. Lotto Texas Select the six winning numbers from 1, 2, , 54.

14. Florida Lotto Select the six winning numbers from 1, 2, , 53.

15. Florida Fantasy 5 Select the five winning numbers from 1, 2, , 36.

16. Wisconsin Badger Five Answer each of the following.

a. Find the probability of selecting the five winning numbers from .

b. The Wisconsin Badger 5 lottery is won by selecting the correct five numbers from. What is the probability of winning if the rules are changed so that in addition to se-

lecting the correct five numbers, you must now select them in the same order as they are drawn?

17. Identity Theft with Social Security Numbers Identity theft often begins by some-one discovering your nine-digit social security number or your credit card number. Answereach of the following. Express probabilities as fractions.

a. What is the probability of randomly generating nine digits and getting your social securitynumber.

b. In the past, many teachers posted grades along with the last four digits of the students so-cial security numbers. If someone already knows the last four digits of your social securitynumber, what is the probability that if they randomly generated the other digits, they wouldmatch yours? Is that something to worry about?

18. Identity Theft with Credit Cards Credit card numbers typically have 16 digits, butnot all of them are random. Answer the following and express probabilities as fractions.

a. What is the probability of randomly generating 16 digits and getting your MasterCardnumber?

b. Receipts often show the last four digits of a credit card number. If those last four digits areknown, what is the probability of randomly generating the other digits of your MasterCardnumber?

c. Discover cards begin with the digits 6011. If you also know the last four digits of a Discovercard, what is the probability of randomly generating the other digits and getting all of themcorrect? Is this something to worry about?

19. Sampling The Bureau of Fisheries once asked for help in finding the shortest route forgetting samples from locations in the Gulf of Mexico. How many routes are possible if sam-ples must be taken at 6 locations from a list of 20 locations?

20. DNA Nucleotides DNA (deoxyribonucleic acid) is made of nucleotides. Each nu-cleotide can contain any one of these nitrogenous bases: A (adenine), G (guanine), C (cyto-sine), T (thymine). If one of those four bases (A, G, C, T) must be selected three times toform a linear triplet, how many different triplets are possible? Note that all four bases can beselected for each of the three components of the triplet.

21. Electricity When testing for current in a cable with five color-coded wires, the authorused a meter to test two wires at a time. How many different tests are required for every possi-ble pairing of two wires?

22. Scheduling Assignments The starting five players for the Boston Celtics basketballteam have agreed to make charity appearances tomorrow night. If you must send three playersto a United Way event and the other two to a Heart Fund event, how many different ways canyou make the assignments?

23. Computer Design In designing a computer, if a byte is defined to be a sequence of8 bits and each bit must be a 0 or 1, how many different bytes are possible? (A byte is oftenused to represent an individual character, such as a letter, digit, or punctuation symbol. Forexample, one coding system represents the letter A as 01000001.) Are there enough differentbytes for the characters that we typically use, such as lower-case letters, capital letters, digits,punctuation symbols, dollar sign, and so on?

1, 2, , 31

1, 2, , 31

1/575,757.

214 Chapter 5 Discrete Probability Distributions

Basic Skills and Concepts

Statistical Literacy and Critical Thinking1. Random Variable What is a random variable? A friend of the author buys one lotteryticket every week in one year. Over the 52 weeks, she counts the number of times that shewon something. In this context, what is the random variable, and what are its possible values?

2. Expected Value A researcher calculates the expected value for the number of girls inthree births. He gets a result of 1.5. He then rounds the result to 2, saying that it is not possi-ble to get 1.5 girls when three babies are born. Is this reasoning correct? Explain.

3. Probability Distribution One of the requirements of a probability distribution is thatthe sum of the probabilities must be 1 (with a small discrepancy allowed for rounding errors).What is the justification for this requirement?

4. Probability Distribution A professional gambler claims that he has loaded a die so thatthe outcomes of 1, 2, 3, 4, 5, 6 have corresponding probabilities of 0.1, 0.2, 0.3, 0.4, 0.5, and0.6. Can he actually do what he has claimed? Is a probability distribution described by listingthe outcomes along with their corresponding probabilities?

Identifying Discrete and Continuous Random Variables. In Exercises 5 and 6,identify the given random variable as being discrete or continuous.5. a. The number of people now driving a car in the United Statesb. The weight of the gold stored in Fort Knoxc. The height of the last airplane that departed from JFK Airport in New York Cityd. The number of cars in San Francisco that crashed last yeare. The time required to fly from Los Angeles to Shanghai

6. a. The total amount (in ounces) of soft drinks that you consumed in the past yearb. The number of cans of soft drinks that you consumed in the past yearc. The number of movies currently playing in U.S. theatersd. The running time of a randomly selected moviee. The cost of making a randomly selected movie

Identifying Probability Distributions. In Exercises 712, determine whether ornot a probability distribution is given. If a probability distribution is given, findits mean and standard deviation. If a probability distribution is not given, iden-tify the requirements that are not satisfied.

5-2

x P (x)

0 0.1251 0.3752 0.3753 0.125

7. Genetic Disorder Three males with an X-linked genetic disor-der have one child each. The random variable x is the number ofchildren among the three who inherit the X-linked genetic disorder.

8. Caffeine Nation In the accompanying table, the randomvariable x